Drag Reduction byPolymer Additives

P. DiamondJ. Harvey

J. KatzD. Nelson

P. Steinhardt

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Drag Reduction By Polymer Additives

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13. ABSTRACT (Muxtmum20Qwor0g

The 1989 JASON Summer Study on Drag Reduction focused on the physics which underliesmethods utilizing polymer studies. The study included a review of drag reductionphenomenology, the development of continuum models of the dynamics of dilute polymersolutions, the introduction of a simple theory of poly-hydrodynamic turbulence and adiscussion of its implications for grid flows, and consideration of optimizing polymerarchitecture.

14. SUBJECT TERMS IS. NUMBER Of PAGES

.polymerized membrances, turbulent flow, dumbell model, 1,. Pcu COoCelastic wave

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Contents

1 OVERVIEW 31.1 Phenomenology of Drag Reduction ............... 41.2 Elementary Facts About Polymers ................ 71.3 Branched Polymers and Polymerized Membranes .......... 10

2 THEORIES OF DRAG REDUCTION 172.1 An Overview of The Problem .................. 172.2 Polymers in Turbulent Flow ................... 192.3 Drag Reduction .......................... 21

3 A MODEL OF TURBULENT POLYMER HYDRODYNAM-ICS 233.1 The Dumbell Model ....................... 233.2 Polymer Hydrodynamic Turbulence ............... 31

4 CONCLUSIONS AND RECOMMENDATIONS 414.1 Conclusions ............................ 414.2 Program Recommendations ................... 42

A BASIC PHENOMENOLOGY OF MHD TURBULENCE 43

iii+.

EXECUTIVE SUMMARY

The 1989 JASON Summer Study on Drag Reduction focused on the

physics which underlies methods utilizing polymer additives. The study in-

cluded a review of drag reduction phenomenology, the development of con-tinuum models of the dynamics of dilute polymer solutions, the introductionof a simple theory of polymer-hydrodynamic turbulence and a discussion

of its implications for grid flows, and consideration of optimizing polymer

architecture.

Current experimental evidence from studies of wall streak spacing andwall stress attenuation suggests that drag reduction occurs by the quench-ing of small scale, high frequency eddys in a buffer layer adjacent to theviscous sub-layer. Theoretical interpretations purport to explain drag reduc-tion in terms of a buffer layer viscosity enhanced by flow-induced polymeruncoiling(G. Ryskin), or in terms of a truncated viscoelastic cascade (P.G.DeGennes and M. Tabor). The uncoiling model is open to question, becauseit is based on the application of intuition from static extensional flows to

fully developed turbulence. The viscoelastic model is incomplete because itfails to consistently account for the ultimate disposition of turbulent eddy

energy.

As a foundation for further theoretical and computational study, a set ofmacroscopic continuum equations which self-consistently describe the fluidand polymer dynamics have been derived. In their simplest form, appro-priate to the linear "dumbell" polymer model, these equations describe t.hedynamics of the polymer extension field and the fluid velocity field, includingback-reaction effects due to the presence of the polymers. The continuum

model is quite similar to magnetohydrodynamics, with the polymer exten-sion field analogous to the magnetic field. In addition to dissipation dueto viscosity and polymer diffusion, a scale-independent damping term, rep-resenting the effects of polymer relaxation modes, appears in the extension

field equation. We have found the analogy between polymer-hydrodynamic

1

turbulence and magnetohydrodynamic turbulence quite useful. The exten-

sion field is excited by fluid eddy stretching, until elastic energy approaches

fluid kinetic energy. This stretching process is limited by polymer relaxation

modes at low frequencies.

When excited, the extensional field inhibits the eddy interaction, resulting

in a change of the inertial range spectrum from 0- to H-I. Direct calcula-tions then indicate the decay rate of grid polymer-hydrodynamic turbulenceexceeds that of ordinary fluid grid turbulence. This is due (primarily) to dis-

sipation due to polymer relaxation modes. Extrapolation to the case of wallflows suggests a picture of a drag-reduced flow as a Newtonian slug with avisco-elastic buffer layer. Enhanced dissipation is due to polymer relaxation

modes.

Consideration was given to optimizing polymer architecture for the pur-pose of drag reduction. While linear chain polymers best meet the dual cri-

teria of low relaxation frequency and high permeated fluid volume fraction

(by weight), branched polymers can better resist degradation by distributing

(mid-coil) flow induced stresses.

2

1 OVERVIEW

Polymer drag reduction was discovered about forty years ago by Toms[l],who observed drag reductions of 30 - 40 percent upon adding only 10ppm

by weight of polymer(methylmethacrylate) to turbulent monochlorobeneze

flowing down a pipe. Similar drag reductions have been observed in poly-

isobutylene polymer in benzene and cyclohexane. For many applications, the

most important solvent is undoubtedly water, for which polyethylene oxide

(PEO) produces very similar effects. Minute concentrations of PEO havebeen used to increase the capacities of irrigation networks, and municipal

sewer systems and to extend the range of fire hoses. The drag reducing

effects of these polymers are most pronounced when the polymers are flex-

ible with high molecular weights, corresponding to N = 104 or N = 10'individual monomer units. Despite the great technological importance of

these materials, there is as yet no fundamental understanding of how this

remarkable drag reduction is achieved. In this section we briefly discuss the

experiments and some important elementary facts about polymer sizes and

relaxation rates. Some novel chemical architectures for polymer additives

are discussed. Section 2 is a critique of existing ideas about polymer drag

reduction. In Section 3.1 we discuss macroscopic continuum models of dilute

solutions of polymers modeled as stretchable dumbells. In Section 3.2, we

discuss the dynamics of polymer-hydrodynamic (PHD) turbulence. The con-

tinuum model presented in Section 3 can be analyzed using methods familiar

from studies of turbulent magneto-hydrodynamics. The analysis suggests in-

teresting and non-trivial modifications of the Kolmogorov turbulence theory.

A prediction of the energy dissipation rate for grid PHD turbulence is given,

and its implications for pipe flow discussed. Section 4 contains our program

recommendations. The basic phenomenology of MHD turbulence is reviewed

in the Appendix.

3

1.1 Phenomenology of Drag Reduction

A typical experimental result[2] is shown in Figure 1-1. For a fluid with

density p and average velocity < v > flowing down a pipe of diameter D the

figure of merit is the mass flux

Q rD2

obtained for a given pressure head Ap. A dimensionless measure of the ratio

of pressure head to mass flux is the "friction factor"

DAp2p < V >21-2)

which is f = 16/Re for laminar flows with Reynolds number

Re = D < v > /p. Although the addition of small amounts of PEO polymer

has no observable effect in the laminar regime, there is a striking change

in the drag above the transition to turbulence which occurs for Re z 2500

obtainable with as little as 5ppm (by weight) of additive. The effect increases

with increasing polymer concentration.

It is well known that enhanced drag in turbulent, high Reynolds number

flows is caused by radial transport of flow momentum by fluid eddys. Stan-

dard pipe flows are well described by the Prandtl mixing length theory[4],

in which the flow is sub-divided into a viscous sublayer near the wall and

an inertial layer extending to the core of the pipe. In the viscous sublayer

(0 < z < v/U., where v is the fluid kinetic viscosity and U. the friction veloc-

ity), the flow profile is linear (U = U2.x/v) and all eddys fall in the dissipation

range. In the inertial layer (v/U. < x), the flow profile follows the logarith-

mic "Law of the Wall" U = (U./Ic)tnx (x is the von Karman constant), with

a spectrum of turbulent eddys of size v31/4/Us.14 < I < x to be found at

each distance x from the wall. At the onset of polymer drag reduction, an

elastic buffer layer adjacent to the viscous sublayer arises. Within the buffer

layer, the flow profile steepens relative to the Newtonian logarithmic profile,

resulting in the formation of a region of "effective slip" (i.e. reduced momen-

tum transport). It is interesting to note that in the buffer layer, fluctuating

4

Polymer concentration (ppm)

A 0 Pure water: 5

0 A 20o 100

*450

Laia f, 0ooI 00 ooO000

10210I01010

Re nod nu0e RAAAD&~R

16 * 00 A A&IAAupe0~~* 0000 AAA AA

U.~* 00AA A

.. 00000 000

102 10 10' 105 106

Reynolds number Re D D<v >/s Re

Figure 1-1. Plot of the "friction factor" defined in the text as a function of Reynolds number forturbulent water flowing in a pipe with various concentrations of PEO. Pointslabelled "N" are with no polymer additives, while points labelled "P" havepolymers in varying concentrations; from Ref. [21

5

radial velocities are smaller than their Newtonian counterparts, but fluctu-ating axial velocities actually increase! However, radial and axial velocitiestend to decouple in drag reduced flows, so that < VU > decreases and thenet turbulent transport of momentum is reduced[3]. Note that the increasein axial fluctuation energy is consistent with the formation of a region of en-hanced slip. As drag reduction increases, the buffer layer expands toward thecenter of the channel while the inertial layer shrinks. It is important to keepin mind that the viscous sublayer remains intact throughout the drag reduc-tion process. Thus, drag reduced flows can be viewed as a Newtonian plug,connected to the viscous sublayer by an elastic buffer layer which supportsenhanced slip.

It is worthwhile to comment on the observations pertaining to the onset ofdrag reduction. Experimental studies indicate that a critical wall shear stressa., is required for the onset of drag reduction, and that ar,, is independent ofpipe diameter, solvent density, or viscosity. A dependence of a,, on polymergyration radius (defined in Section 1.2) RG of the form a',,, RGja is observed,with 2 < a < 3. The gyration radius RG is virtually always significantlysmaller than the dissipation scales of turbulent flows of Reynolds numbersRe < 106.

There are also experiments[5] which monitor the friction factor as a func-

tion of the distance downstream from a point in the center of the pipe wherethe drag reducing agent is injected. Some drag reduction appears even be-

fore significant amounts of additive have had time to diffuse to the wall. One

(controversial) interpretation of these experiments is that polymers do not

merely lubricate the boundary layer at the wall but also act to modify the

approximately homogeneous isotropic turbulence in the center of the pipe.We are unaware of experiments on changes induced by polymers added to

grid-generated turbulence, away from walls, which could clarify this point.

6

1.2 Elementary Facts About Polymers

To get some feeling for the ways in which polymer additives might affect

turbulent flow, we need to estimate their characteristic size and time scale[6].

A simple random walk model of a polymer chain with N monomer units,which ignores the interactions between different monomers, predicts a size or"radius of gyration" R. which grows like NY',

R, FsaNI , (1-3)

where a is the monomer size. A more accurate formula, which takes intoaccount swelling due to self-avoiding interactions between monomers is,

.9 ;. aN5. (1-4)

The chemical formula for polyethylene oxide with N monomers is

... I-CH2 - CH2 - O--]N .... (1-5)

Each monomer unit is about a ; 5A. The density of polymeric material with

N = 10i inside a cube with edge R. centered on this very tenuous object isthus only 4 x 10-4g/cm3 !. This remarkably low density of monomer means

that no water molecule is ever very far from a flexible polymer chain even

at very low concentrations of chains. The volume, for example, enclosed by

the radii of gyration of 1.4 x 10i' PEO molecules with N = 10i added toone cubic centimeter of water (corresponding to a concentration of 100ppm

by weight) is about 25 percent of the available volume.

Each polymer chain is a delicate little mechanical transponder, with itsown set of internal relaxation times, which reacts to the stresses associated

with fluid motion. Although the spectrum of relaxation times of a polymer

in a solvent is complicated, the slowest dynamical mode is associated withdistortiuns of the (approximately spherical) equilibrium polymer shape and

is known to be[6]

'r. N's (1-6)kBT

7

where q/, is the solvent viscosity. The important time scale r. is called the

"Zimm time" and is of order milliseconds for PEO with N = 10i monomer

units. To within numerical factors, it can be written as[6]

r. 1,/ 71K(1-7)

i.e., as the ratio of the Stokes drag coefficient y = 67r77R, for a slightly

stretched poly.aer of characteristic size R. and a Hookean force constant

K = kBTIR2. Temperature appears in the effective polymer spring constant

because the restoring force for a stretched polymer is entropic in origin, due tothe many more configurations available to a polymer in its coiled, as opposed

to stretched state.

When polymers interfere with turbulence, it is because they distort inthe fluctuating velocity field and eventually react back on the flow. The

approximately 1200A size of a typical polymer is much smaller than the

Kolmogorov inner scale for most turbulent flows. Thus the polymers see aneffectively uniform random straining field from all the eddies in the cascade.

The Zimm time determines which eddies will have a strong effect on thepolymers since they will quickly reassume their equilibrium shape in response

to strains induced by an eddy of size I unless the eddy turnover time is shorter

than r7,. The strain rate j(1) (i.e., the inverse eddy turnover time) for an eddyof size I within the Kolmogorov theory is well-known to be

-j(l) = EI/l1. (1-8)

The only eddies in the turbulent cascade which will be able to distort the

polymer shape are those for which[7][8]

()>1, (1 -

i.e., those eddies with 1 < P' where

r = (vr,)~ eI N2-7. (1 - 10)

It is appropriate to inteject here that the observation that a critical

wall Reynolds stress must be exceeded for drag reduction to result is closely

8

related to the time criterion expressed by Equation (1-9). This is easily seen

by noting that for wall flows, the maximal eddy shearing rate is

j = U2./v = U./lx,, where xz, = v/U. is the width of the viscous sublayer.

Thus, drag reduction will occur if U.1/x, > Tr1 , which is equivalent to U2 >

v/r,. This is equivalent to the condition that

r, = pU.2 > kBT/R 9, (1 - 11)

which defines a critical wall Reynolds stress varying inversely with R andwhich is independent of solvent density, viscosity and pipe radius.

The length scale 1* is independent of the polymer concentration, and is

of order 10- 2cm for a typical C : 106 cm 2/sec3 . Possible consequences of thepolymer distortions induced by eddies smaller than this scale but larger than

the Kolmogorov inner scale Id will be discussed in Sections 2 and 3. Here, we

simply note that strongly stretched polymers will begin to react back on the

flow on those scales 1, ld < I < l*, when the restoring forces per unit volume

are comparable to the Reynold's stresses in the turbulent fluid[8]. For "ideal"

polymers, whose equilibrium size is given by Equation (1-3), the restoring

force f for a single polymer is linear in the distorted polymer size Q[6],

f = KQ = kBTQ/R'. (1-12)

Here, K is the Hookean spring constant discussed above, and Q, (see Section

3) can be defined in terms of a hydrodynamic average of a stretching vector

describing the distortion of each polymer. Note that the energy associated

with this force, E = 1KQ 2, is comparable to the thermal energy kET when

Q = Rg. This result applies only for weak distortions of polymers. The forcewill be much larger when the polymer is fully extended, i.e., for Q ,., Na.

When excluded volume interactions are taken into account, there is also an

intermediate regime of stretching, R, << Q << Na, for which[6]

f ; kBTQ2/1R. (1 - 13)

A complete theory of polymer drag reduction must determine how the non-

linear springs embodied in Equation (1-13) alter the flow which stretch them.The construction of such a theory is a formidable challenge; as we shall see

in Section 3, the hydrodynamics of turbulent flows in which the polymers

behave as linear, Hookean springs is already quite nontrivial.

9

1.3 Branched Polymers and Polymerized Membranes

The simple ideas sketched above allow us to estimate whether the syn-thesis of polymers with novel, nonlinear architectures would be useful inturbulent drag reduction. It is possible to synthesize branched polymers,for example, whose connectivity is illustrated in Figure 1-2. There is cur-rently considerable interest in the other novel architecture shown in Figure1-2, that of polymerized membranes[9]. These flexible "tethered surfaces"assume configurations in a solvent not unlike crumpled pieces of paper.

Upon first consideration, branched polymers and polymerized membranesappear to be disadvantageous compared with linear polymers. The key rela-tion is the scaling of the radius of gyration, Rg, with the number of monomerunits, N:

Rg Pe AM; (1-14)

here n is the number of nodes in a branched polymer or tethered surface,and a' is proportional to m3'/ where m is the average number of monomersper nodes (inn ; N). The exponent, v, is approximately, 0.5 for branchedpolymers[10], and 0.4 for polymerized surfaces[9].

For fixed N, the radius of gyration decreases as the number of nodesincreases, which makes the polymer less effective in drag reduction. Onewishes to increase the Zimm time as much as possible so as to 'activate' thepolymers by eddy-induced stretching at increasingly larger onset scales givenby (see Equation (1-10)), e2 , Ci/ 2t 9/ 2 /(kBT)3/2. Yet, the Zimm time isproportional to R3, and hence is shorter (for fixed N) for branched polymersor tethered surfaces compared to linear polymers. Second, one wants topermeate a significant fraction of the fluid with polymer even though theyare a negligible component by weight. The relevant figure of merit is themass density of monomer inside the radius of gyration.

ýLo~(1 15) N-

where M0 is the monomer molecular weight and N. is Avogadro's number.For a fixed number of monomers, N, the cheapest additives will be those

10

(a) (b)

(C)

Figure 1-2. Various chemical linkages which could be used for turbulent drag reduction: (a)conventional linear polymer; (b) Branched polymer characterized by a fixed ratiouf branch points to monomer units; (c) A polymerized membrane, with theinternal connectivity of a flat two dimensional surface. Such surfaces can be mademore flexible by connecting the nodes with polymer chains, as in a twodimensional gel.

11

which produce the most tenuous structures, i.e., those with the lowest valueof p or the largest v. Not surprisingly, linear polymers are also most efficientfrom this point of view.

However, branched polymers or tethered surfaces have the potential ad-vantage that they will be less susceptible to degradation at high Reynoldsnumbers. When linear polymers are fully stretched in the extensional flowfields, they break (usually near the center) and become less effective in reduc-ing drag. To appreciate this effect, let us model the stretched linear polymeras a set of spheres of radius b connected by springs with spring constant k,as shown in Figure 1-3. We will assume that the polymer is stretched lin-early along the x-direction by an extensional flow field, v- = "jx, as might becreated locally by turbulent eddies, and that the flow field is centered on themiddle of the polymer. If only the spheres interact hydrodynamically withthe flow field, the net force on the nth sphere is

Fn = k8._ 1 - k6b - 67r-qbjxn, (1- 16)

where n = 0 is the center of the polymer, xn is the position of the nth sphere,6. =_ z+, - Zx - - is the extension of the nth spring from its equilibriumlength 2, and p is the viscosity of the fluid. In equilibrium, F = 0. Oneimmediately observes that 6n-1 > 6n; in other words, springs closer to thecenter of the polymer are stretched more. If we take the continuum limit,replacing n by the continuous variable s, Equation (1-16) becomes

d6Cs) = -x(S), (1 -17)ds

where a = 6ftqlbj/k. Near the center of the spring, ds oc dx, so Equation (1-16) has the solution 6 - (a/2)(L2 - x 2), indicating a parabolic falloff in thetension as one moves away from the center of the polymer. This argumentmay explain why experiments on polymer degradation at ultrahigh strainrates indicate breaking near the middle of the chain.

The branched polymer illustrated in Figure 1-3 (or a tethered surface)may alleviate the degradation problem. Let us first suppose that the branchedpolymer is arranged so that the end with fewer branches is furthest away from

12

XN-I XN XN+I

o oo

Extensional K6N-

N

Flow Field

(a)

0 0 XN-i

0 0 0 X XN X N+1

0 0 0 -- ----go KS N

o 0 0 -Z bE :o - F o

Figure 1-3. Polymer ball-and-spring model in a high Reynolds number extensional flowfield, v = -x. The center of the polymer is at x = 0 and all springs arestretched nearby along the x-axis. FS is the stoker drag on the Nth ball,Fs = 6ri7b-;N. [(a) Linear polymer (b) Branch polymer.]

13

the center of the extensional flow field (see remarks below). The tension in-creases as one proceeds inwards from the end of the polymer until one reaches

a node. At the node, there are two (or more) polymer strands which sharethe load; that is, the coefficient of the first term in Equation (1-15) is multi-

plied by two (or more), so now b.-. may be less than 6b. Adding more nodes

as one proceeds closer to the center of the flow field continues to limit thetension, and, hence, avoids degradation at higher Reynolds numbers. Giventhat the radius of gyration decreases as the number of nodes increases, onewishes to have the minimum number of nodes necessary to avoid degradation

at the design Reynolds number.

In simple models, branched polymers begin from linear chains that growout from a central core and then bifurcate after some fixed average numberof monomer units; the branches then bifurcate after growing and extendinga similar number of monomer units, etc. If such a branched polymer were

placed in an extensional flow field with the central node in the middle, theeffect would be the opposite of that described above. The tension on themonomers in the many distant limbs would be focused on the central node.This may not be a real problem since this would merely cause the originalbranches to snap from the central node, in which case each fragment wouldbe similar to the cartoon in Figure 1-3. However, it remains unclear whetherthe fragments will align themselves in the flow field so that the branch thatwas central originally moves to the farthest part in the flow field.

Polymerized surfaces may be better candidates in that their behavior ismore predictable. We might envisage the surface as a square grid of polymerchains meeting at nodes. In an extensional flow field, the grid will deforminto an extended, nearly linear configuration with the comers, say, furthestfrom the center of the extensional flow field. As with the branched polymers,the nodes in the grid can limit the tension on segments near the center of theflow field. The disadvantage of polymerized surfaces compared to branchedpolymers is that v is smaller. However, if the grid is rather sparse (fewnodes), this may not be so significant.

In conclusion, it appears that the application of polymers with novel

14

architectures is a possible mechanism for extending polymer drag reduction

to flows with very high Reynolds number. The addition of crosslinks helps

the ploymers resist degradation, but also means that greater concentrations

of polymers are necessary to obtain the same drag reduction.

15

2 THEORIES OF DRAG REDUCTION

2.1 An Overview of The Problem

Drag reduction by polymers in turbulent flows is an extremely compli-

cated problem. It combines the complexity of turbulent flow (difficult even

for a Newtonian fluid) with the problems of polymer physics; their combina-tion changes the character of the turbulence and leads to a yet more complex

and difficult problem.

A naive attempt to understand the effects of polymers on drag could goastray in any one of several ways. Dissolved polymers increase the viscos-

ity of polymer solutions, particularly if the polymer molecules are extended;it might therefore be (incorrectly!) inferred that they would increase drag(albeit slightly), just as would an increase in the viscosity of a pure New-tonian solvent, even at high Reynolds numbers. The fact that in turbulent

flows the observed effect is in the opposite direction is sufficient to establish

the subtlety of the physics involved. So long as the flow remains laminar,

the naive drag increase is expected to be correct; however in laminar flowthe polymers usually remain coiled so this effect would be very small. Itinstead might be supposed that polymers could have no effect at all on drag,

because the rate at which viscous work is done is determined, in a stan-

dard Kolmogorov cascade picture, by the largest scale velocities of the flow.However, polymer-enhanced viscous stresses are significant only on the small

scales; which satisfy the Zimm time criterion w, < V(t)/L. Larger eddys, forwhich V(1)/t < w,, are unaffected by the presence of polymers. However it

must be noted that momentum transport in bounded shear flows is critically

sensitive to the dynamics of such small scale eddys immediately outside the

viscous sublayer.

It is evident from the data (Figure 1-1) that polymer additives do notaffect the onset of turbulence. This is probably due to the fact that the un-

stable sheared flows which commonly preceed the onset of turbulence rotate

17

polymer chains but do not stretch them. Their effect is felt in turbulentflow, which, for these data, occurs when Re > 2500. At such high Reynoldsnumbers turbulence is expected to be fully developed, and manifest a cas-cade of turbulent energy. Yet, as just discussed, the properties of such acascade (except for the location in k-space of its inner scale) are independentof the magnitude of the viscosity. These properties include the dissipationrate, which is determined by the flow at the outer (largest) scale, and whosevolume average is proportional (by conservation of energy) to the global flowdrag or friction factor.

The (Kolmogorov) cascade model must therefore be modified for polymersolutions. One possible modification is simply in the numerical coefficients.The Kolmogorov analysis is entirely dimensional, and cannot determine thecoefficients. It is plausible that the non-Newtonian, anisotropic, and hys-teretic nature of the viscosity of a polymer solution changes the effectivevalues of the coefficients. This possibility can be tested by detailed numeri-cal simulation of turbulence in such fluids.

A second possibility is that the spatial variation in the effective viscos-ity of polymer solution pipe flow is essential. In this scenario, advanced byLumley[7], polymers in the boundary layer are thought to be stretched andextended by inertial range eddys! As a result of the ensuing coil-stretch tran-sition, the viscosity increases dramatically. It is important to keep in mindthat this buffer-layer region of enhanced viscosity is located immediately nextto the viscous sublayer, and extends to a distance from the wall x = U.1W',thus encompassing the eddys that satisfy the Zimm criterion. As a resultof the locally increased viscosity, transport of stream-wise momentum to thewall decreases (due to small scale eddy damping) and drag is thus reduced.

A third possibility is that the Kolmogorov cascade is truncated, or iseven inapplicable because energy propagates up the cascade (to longer wave-lengths) instead of down it, violating Kolmogorov's central assumption.

An important question not directly addressed by experiments to date iswhether polymer extension is required for drag reduction. Most, but not

18

all, theories would predict that this is so. The prediction could be tested inexperiments with sufficiently concentration solutions of short-chain polymers.At the onset of turbulence the strain rates will be insufficient for them toundergo the coil-stretch transition, and most theories will predict little dragreduction. At high Reynolds numbers and higher turbulent strain rates, thereshould be an onset (sudden or gradual) of drag reduction.

2.2 Polymers in Turbulent Flow

The behavior of dissolved polymers in flow fields has been the subject ofan extensive literature. For example, Henyey and Rabin[1 1] have calculatedequilibrium polymer conformations in stationary flow fields. The problem ofdynamic polymer relaxation to an equilibrium state in specified stationaryflow fields is considerably more complex (Rabin, Henyey and Pathria)[12];Rabin[13]. The problem of polymers in turbulent flow fields is yet more diffi-cult: the flow field is varying spatially and temporally, and the quantitativenature of these variations can be specified only statistically (and even thisspecification is both difficult and uncertain).

There are a number of models and an extensive literature on the subjectof dissolved polymer response to flow fields. Two simple limiting modelsare the dumbell model (for example, as developed in a later chapter of thisreport) and the affine deformation model, in which the polymer is passivelydeformed following the strain tensor of the embedding fluid. It is frequentlyargued that a great degree of polymer extension (necessary in order that alow polymer concentration produce significant effects on the flow) requiresthat some component of the strain rate tensor exceed the Zimm relaxationrate of the polymer, proportional to N-1 8 in a good solvent, where N isthe number of monomers in the polymer chain. This argument has beenchallenged by RHP[12] who present a free energy argument which predictsa dependence proportional to N-1 '6 in good solvents. They assert that thislatter form is in better agreement with the data. Both these arguments areessentially kinetic, in that they depend on overcoming the entropic barrier

19

to full polymer extension; in any significant stationary extensional flow field

the equilibrium state of a long polymer is nearly fully extended. This follows

from the Kramers potential describing the tendency of the viscous drag force

to distend the polymer, which varies as RJ, a steeper dependence than the

entropic forces which tend to keep it coiled. For small extensions (coiled

polymers) R is small and the entropic forces dominate. The arguments differ

in that the RHP argument demands that there be no entropic barrier, while

the Zimm time argument demands only that the entropic barrier be ineffec-

tive (according to an approximate argument based on relaxation rates). One

might question whether there can be anything like this sharp phase transition

to an elongated state when the flow is highly turbulent; phase transitions are

often smeared out in systems subjected to chaotic, time-dependent forces[6].

The problem of polymer extension in turbulent flow fields is particularly

difficult and subtle. The principal axes of extension are changing with a

characteristic decorrelation time rd. The flow has a fluctuating vorticity of

characteristic magnitude w, which rotates the polymers, and tends to inter-

fere with their extension (it is for this reason that relatively little extension is

obtained in pure shear flow, for which the magnitude of vorticity equals the

largest component of the rate of strain tensor). Of course, for Kolmogorov

turbulence, w -r;"1. At high strain rates a polymer molecule will have neg-

ligible entropic restoring forces, and will passively (affinely) follow the flow.

In a stationary purely extensional flow field with the rate of strain &y/Or its

length will increase

In the turbulent flow field, after a single decorrelation time the extension will

be

£ toexp ( ,Td) (2-2)

where the effects of vorticity are qualitatively included by defining rd in axes

and along a path which follow the polymer molecule.

It is evident that the polymer extension depends very sensitively on

8VT- rd. (2-3)

20

As noted above, this quantity may be estimated, on dimensional grounds, to

be of order unity for a turbulent flow. It is clear that its actual quantitative

value (or rather, that of the analogous quantity in a more quantitative the-ory) is of great importance, and is a sensitive measure of the nature of the

turbulent flow. However, it does appear unlikely that significant extension is

possible in highly turbulent flows.

2.3 Drag Reduction

Very little of the extensive literature on polymers in turbulent flow actu-ally attempts to explain why there should be a reduction in drag. The classicwork of Lumley[7] argues that if polymer molecules are extended by turbu-

lence in pipe flow, they lead to a flow in which two definitions of the laminarboundary thickness are possible - one based on the viscosity with unextendedpolymers, which should be applicable in the laminar viscous sublayer, anda larger value calculated from the viscosity in the turbulent region with ex-tended polymers. He argues that between these two distances from the wallthere should be a buffer layer into which the turbulence has difficulty pene-trating, and thus through which momentum transfer is slow, thus reducingthe wall drag. This buffer layer corresponds to the region of enhanced slip

(i.e. good strearnwise momentum confinement). This model is qualitatively

reasonable, but is not quantitative enough to test.

Ryskin[14] describes a "yo-yo" model for affine polymer deformation inextensional flow. It appears to satisfactorily describe convergent flow in acone. When applied to pipe flow, this model resembles Lumley's; its treat-

ment of polymer dynamics in a specified flow field is quantitative, but itsapplication to turbulence and its treatment of drag reduction are partly phe-

nomenology, and partly semi-empirical fit to the data.

Tabor and de Gennes[15] (see also de Gennes[8] for more detail) presenta very different model. They assume affine polymer deformation. In orderto calculate the degree of extension, they assume that the fluid strain is a

unique function of the length scale of the turbulence, in analogy to laminar

21

cone flow in n dimensions, where n is a non-integral number between 1 and

2. Each polymer molecule is assumed to be affected only be eddies of a

single size. This description is peculiar and of dubious applicability because

cone flow is strongly convergent and has a large (negative) divergence at

its apex, while incompressible pipe flow and turbulence are divergence-free.

Also, a given molecule is simultaneously subjected to turbulent eddies ofall sizes, as pointed out by Ryskin[14]. Having made these assumptions,they define a new inner scale t1* for the turbulent flow field, by equatingthe polymer elastic energy density to the Reynolds stress of the turbulence.By analogy with the early work of Lumley, but without further quantitativedetail, they argue that the existence of this larger inner scale for the polymer- hydrodynamic turbulent flow leads to drag reduction by truncationi of theKolmogorov cascade.

The presence of extended polymers in a turbulent flow will increase theviscous dissipation in eddies of a given size, and thus increase the inner scalesize of the flow. It is possible to construct a model analogous to that ofTabor and de Gennes in which the new inner scale is set by equating theviscous dissipation rate in flow about the polymer molecules to the rate ofdissipation irn the turbulence:

U(P ) cv'R/ (2-4)

where cp is the number density of polymers, R their radius of gyration, 7the shear rate, ii the solvent viscosity and p its density, and U(r) is thecharacteristic velocity of eddies of size r. Following Tabor and de Gennes wewrite

R = & (2-5)

A (2-6)

where t* is the length scale at which i(t1) equals the Zimm relaxation rate(the criterion for afline polymer deformation). Assuming Kolmogorov turbu-

lence with dissipation rate e and estimating i(r) = U(r)/r, a little algebraleads to

22(2-

22

3 A MODEL OF TURBULENT POLYMER

HYDRODYNAMICS

3.1 The Dumbell Model

We focus our attention on the two end points of a long polymer chain, and

replace its many internal degrees of freedom by a stretchable dumbell which

spans the points-see Figure 3-1. Although clearly a gross oversimplification,

this crude model has proven useful in studies of laminar polymeric flow[16],

and leads to nontrivial modifications of homogeneous isotropic turbulence.

The equations of motion of the two endpoints are assumed to beS) ou (3-1

7 ,dt - Tf-" -0- ' I 3 1

(dF_. = 0r

T -V2 --- +(2,,

where 61 and i'2 are the fluid velocities at the endpoints, - s V,.R. is the

Stoke's drag coefficient for the part of the pol--i-er represented by the end-

point, and the spring potential is

(', = F -KIf, -F12 (3-2)

with K = kBT/R,. Although nonlinear spring potentials like that repre-

sented by Equation (1-13) are more realistic in the highly stretched regime,

we shall for simplicity confine our attention to the more tractable linear case.

In setting 7 - i, R9 , we make the usual assumption[4] that the fluid in the

vicinity of the polymer chain is dragged along by the chain motion, so that

the drag is of order the Stokes force on an object cf size Rg. The functions

((t) and C2(t) are thermal noise sources, with autocorrelation functions

< Ci(t)Q(t') >= 2ykT6ij6(t - t'), (3- 3)

which serve to equilibrate the dumbell when there is no macroscopic fluid

motion.

23

'4-.

.-.

Figure 3-1. Long chain polymer with endpoints r, and r2 modeled as a stretchable dumbbell.

24

We imagine that the fluid is filled with a dilute solution of these dumbells

and pass to center-of-mass and relative coordinates for the two ends,

1 r+ F2 (3-4)=q' ri - rb,

in terms of which the Langevin Equation (3-1) may be written

dR = i+ 2-f[ 1(t) + (2(t)] (3-5)

df 20OU 1S= W . V ) 7 q + [CI(t) - C2(t)].

Here i' is the external velocity field evaluated at the center-of-mass position,

and we have expanded i' - -2 in gradients of the velocity field at this point

to get Equation (3-5). Standard methods[17] then lead to a Fokker-Planckequation for the distribution function f((A, q, t) describing the midpoints andend-to-end distances of the ensemble of dumbells, namely

Of a [-- D. a (3-6)

at - R vo aRj

0 [(F . 2Ua) f]- o-?" q"V)v--- oD€ -)

where f is normalized so that

Jd3Rfd3qf(Aq,,t) = 1. (3-7)

The diffusion constants D. and Dq arise from the Langevin noise sources,

and are given by

Do = kBT/27 (3-8)

Dq = 2kBT/7.

In a steady homogeneous shear flow parametized by the constant matrix

C' Civj -ai, (3 -9)

one can check that

f,(f(, q") = exp [-(u(q')/kBT) + %ijqiqj] (3 - 10)

25

is a steady state solution of (3-6).

Hydrodynamic equations follow from taking various moments of Equation

(3-6). The concentration c(r', t) of dumbells

c(r, t) = d3qf(F, q, t) (3-11)

for example, obeys the equationO~c + (t7. )c = D.V'c. (3-12)

An initially inhomogeneous dumbell distribution subjected to homogeneous

isotropic turbulence will be mixed to uniformity in a few eddy turnover times.

Henceforth, we neglect the complication of inhomogeneities in the polymer

concentration, and simply set

c(f, t) =- cp -,, const. (3 - 13)

It is also illuminating to take the first moment of Equation (3-6) and consider

the dynamics of the average end-to-end distance

V(t) = J d3qff(rf, f, t) (3 - 14)

which satisfies

, -I- (V. V)O ( V)V_ 2W, -+- DJV2, (3-15)

with w, = K/-y. In addition to the usual streaming and diffusive terms,

there is a line stretching term (0 . V)v. This tendency of velocity gradients

to stretch the dumbells is resisted by the term -2wzQ, where w, is of order

the reciprocal of the Zimm relaxation time discussed in Section 1.0.

Equation (3-15) is in fact inappropriate for the problem at hand; to see

this, note first that, as shown in Figure 3-2, many different polymers will

interact with a given volume of fluid. Any attempt to define a hydrodynamic

field Q(r, t) by averaging polymer stretching vectors over this volume will

depend on purely arbitrary conventions such as whether we set q" = r', -

or q = r2 - rf for a given dumbell. If one eliminates the ambiguity by, say,

using polymers which are deuterated at one end, the configuration q" will be

26

II;

Figure 3-2. Elongated dumbbells in a hydrodynamic averaging volume, together with themicroscopic stretching vectors {' a). The appropriate order parameterdescribing the state of alignment is QU = <qaiqaj>. The sense of the arrowsdepends on a arbitrary convention, which drops out with this definition ofthe order parameter.

27

as likely as -q', and the hydrodynamic average Q(', t) will be zero, even if

the stretched polymers are aligned! The same problem appears in nematicliquid crystals, which are fluids of aligned rod-shaped molecules with inver-sion symmetry[18]. As in the case of nematics the appropriate hydrodynamic

field is a tensor,

j( = J dqqiqjf(rqqt), (3- 16)

which is manifestly invariant under q' - -q' The equation of motion which

follows from Equation (3-6) is

OAQij + (V" V')Qi = QiyOyvj + Qi, &,vj - 4w'Qij + DV 2Qj + 4kB-Tb1,•.7(3 - 17)

In addition to streaming, stretching and diffusive terms similar to those ap-pearing in Equation (3-15), there is a thermal forcing contribution (4kBT/7 )6Sj.This term balances against the Zimm relaxation term -2w.Qij in a uniformquiescent fluid to give the expected equilibrium solution of (3-17), namely

Qi = ik R26 3 (3- 18)

corresponding to an isotropic distribution of polymers with size R,.

The tensor Qij(rt) determines how the stretched polymer distributionreacts back on the fluiO. Taking over the analysis for laminar polymeric flow

in Reference [16], we find that the equation of motion for the (incompressible,

V " V= 0) velocity field is

P [Li + (". 'vi] = -V"p + c + 1iV 2v,. (3-19)

The new term, cpKOjQj, is proportional to the concentration of polymersand represents a contribution to the fluid stress tensor of the form

oij = cpKQj. (3 - 20)

As explained in [16], it arises from the forces exerted on the fluid by dumbellswhich pierce the walls of the hydrodynamic averaging volume.

Equations (3-17) and (3-19) comprise a nontrivial hydrodynamnical modelof polymeric fluids which would be very interesting to study at high Reynold's

28

numbers via computer simulations. Except for its tensorial character (Q,, has

six independent components), Equation (3-17) is like the equation which de-

scribes the stretching of the magnetic field in magnetohydrodynamics (MHD)[19].

We expect that numerical codes developed for MHD could be readily adapted

to study polymer hydrodynamics (PHD!). Although nonlinear dumbell springs

are easily incorpnrated into the velocity equation, such nonlinearities lead to

an infinite hierarchy of equations for moments involving ', which are less

straightforward to simulate. When Qj x bij, as in Equation (3-18), the

polymeric effects can be incorporated into an additional (osmotic) contribu-tion to the total pressure. More generally, however, we must decompose Qj1into diagonal and traceless parts,

Q,,(f, t) = Q('1, t)6,j + Qij(, t) (3 - 21)

where TrQij = 0, and observe that Qij contributes in a nontrivial way to thedynamics of an incompressible fluid. The isotropic and traceless parts of Qjare coupled together in Equation (3-17) by the stretching terms.

Some insight into the behavior of turbulent polymeric fluids follows fromthe observation that, in the absence of dissipation and thermal forcing, only

the sum of the fluid kinetic energy and polymer potential energy is conserved.Indeed, it is straightforward to show from Equations (3-17) and (3-19) that

ifI J dar[pv2 + cpKQkk], (3 - 22)ETOT (3 22

then"dETOT - d3r[pv(Oivj)2 + 4wzQkk] (3 - 23)

where ear is the volume of the sample. We imagine that the macroscopic

forcing of the kinetic degrees of freedom has been turned off and ask how theenergy decays. Under circumstances such that the polymers are only slightly

stretched,2 cpKQkk is a negligible fraction of the kinetic energy and theconventional Kolmogorov cascade picture should be at least approximatelycorrect. We argued, however, in Section 1.0 that eddies with characteristicsize I < V - (e/w,)I would produce effectively uniform random shearswhich would strongly distort the polymers. The same criterion emerges fromcomparing the Kolmogorov estimate of the stretching matrix ( 1iv,) with

29

the Zimm rate -w, on the right-hand side of Equation (3-17). Because oftheir coiled equilibrium state, polymers are capable of enormous extensionsand their potential energy can become a significant fraction of the totalenergy in the flow. One "viscoelastic" scenario[8],[15] is that kinetic andpotential energy are exchanged between the fluid and polymer degrees offreedom over a range of scales td < t < te" < t* in a way which effectivelytruncates the propagation of the kinetic energy down to the Kolmogorovinner scale. Energy is then destroyed at a slower rate by the dissipativeterms on the right-hand side of Equation (3-23), because the average valueof pV(8,v,) 2 decreases. If this effect of adding polymers is larger than theincreased dissipation caused by the Zimm term wZQkk, energy will decay ata slower rate and the drag will go down.

Tests and elaborations of these ideas via spectral closure methods anddirect numerical simulations would clearly be very desirable. The scenario ofTabor and DeGennes described in Section 2.0, in particular, could be testedexplicitly. The "viscoelastic" range of length scales they propose bears somesimilarity to a range of energy equipartition between magnetic and kineticdegrees of freedom which occurs in models of turbulent MHD, mediated byAlfven waves[20]. Equations (3-17) and (3-19) do, in fact, admit "polymerwave" solutions which are similar to Alfven waves. To see this, we firstimagine that random stretching by the turbulent velocity field on scales t < t,has momentarily aligned the dumbells, as in Figure 1-3. This is like thegeneration of a local region of intense magnetic field in MHD. We then lookfor excitations out of this state of alignment which couple to the velocityfield. We write the tensor Qj as

Qij s; (Q', + 6Qi)(Qoj + 6Qj) (3-24)

st Q.,Q. + Q~i6Q, + 6QiQQ,

with . 6bd., and look for normal modes of (3-17) and (3-19) of the form

d = 6bQ 0e"''-''t (3-25)

iT = oe

with 6( . • = 0 = 0. For wavevectors k such that 1,'1 < k < (t d)-1 , wecan neglect both the viscous damping and the relaxation terms in Equation

30

(3-17) and find the characteristic frequency

w(k) = • o(0o,. j)2. (3-26)

We have assumed that the local alignment produces a Q. which greatlyexceeds R,, so that the forcing term in Equation (3-17) can also be neglected.

For the above calculation to be meaningful, the frequency Equation (3-26)must be higher that the inverse turnover time of the eddy which producedthe region of alignment, i.e., we require

w(k) > 0k. (3-27)

In the region of wavevectors where this happens, kinetic energy will convertinto potential energy during the lifetime of the aligned region, and the con-ventional Kolmogorov cascade will be modified or terminated. To calculatewhere this region occurs, we must evaluate Equation (3-26), and hence es-timate the root mean square stretching Q. on a scale I - k-1 due to allrelevant larger scales. Following heuristic methods developed for MHD tur-bulence[20], we write this as

(k) 14T dkk2~ji(k) (3 -28)

where

S= J d3reirQ1ii(f) (3 - 29)

and where the lower limit arises because scales I > 4, have no effect on thepolymers. Knowing the function Q.(k) is equivalent to knowing the exponentn in Equation (2-8). Simulations of Equations (3-17) and (3-19) would helptest this picture, and determine the exponent n.

3.2 Polymer Hydrodynamic Turbulence

Having described the dumbell model of polymer hydrodynamics, we nowturn to a discussion of the dynamics of turbulent polymer solutions. Whilebounded pipe and channel flows are of ultimate interest, we here focus on

31

the simpler (yet quite formidable) problem of understanding the physics ofhomogeneous polymer hydrodynamic turbulence produced by the flow of thesolution thru a fine grid (hereafter referred to as grid turbulence). Our ap-proach is to exploit the formal analogy between polymer hydrodynamics andmagnetohydrodynamic turbulence, the essential physics of which is summa-rized in Appendix A. As noted previously, the polymer hydrodynamic ex-tension field Q is analogous to the magnetic field B, so that polymer elasticwaves, which propagate at VE = (CPK Q2/po)l/2 are the formal counter-

part of Alfven waves, which propagate at VA = (B.2/p) 1 /2. An importantdifference between the two systems occurs in the scale dependence of thedissipation. In the case of magnetohydrodynamics, magnetic energy is dissi-pated by resistive diffusion, so that ,k2 characterizes the damping rate of Bk.In contrast, dissipation of elastic energy occurs at the Zimm relaxation ratew,, which is scale independent. Hence, while the magnetic Reynolds numberRM(e) = IV(t)/rl decreases with eddy scale size 1, the analogous 'elastic'Reynolds number RE(e) = V(e)/ltw actually increases as I decreases, untilthe rather feeble effects of polymer-concentration diffusivity assert themselvesat microscopic scales (i.e. typically D. < v, where v is the fluid kinematicviscosity). Hence, dissipative effects associated with polymer dynamics aremost important for scales which marginally satisfy the Zimm criterion (i.e.scales I such that V(s)/It -, w.), and decrease in importance throughout theremainder of the inertial range.

In the production of grid turbulence, solution flows thru a hot wire gridof mesh spacing L at velocity V.. Turbulence then evolves (rapidly) as thefluid flows downstream of the grid (i.e. see Figure 3-3). The downstreamevolution of polymer hydrodynamic turbulence may be considered to occurin two sequential stages:

1. the elastization phase[21],[22],[23] - a transient phase during which theheretofore hydrodynamically 'passive' polymer coils are stretched, sothat some of the mechanical energy of inertial range eddys is convertedinto polymer elastic energy.

32

In-Flow Grid Turbulence

L

Figure 3-3. The Production and Evolution of Grid Turbulence.

33

2. the viscoelastic turbulence phase[20],[24] - the stage of fully developed

viscoelastic turbulence during which the coupling of hydrodynamic ed-dys to elastic waves modifies the spectrum and energy dissipation rate

of the turbulence. Throughout this discussion, it is important to keep

in mind that the turbulence is forced at the constant rate c - V,3/L.

Hence, e should be thought of as a forcing rate, rather than as a "dis-sipation rate," its usual meaning. The actual dissipation rate will be

computed usingdETOT rdEtO J d3r{pv(! V + 2w2 Q2}" (3-30)

Also, it is understood that the turbulence dynamics are modified onlyon scales for which the Zimm criterion is satisfied (i.e. for which V(f)/f >

W.).

In the elastization phase, passive, initially coiled polymers are stretched

by turbulent eddys. Since

dQ-= = 2= .VV-w (3-31)

it follows that the rate of polymer stretching due to eddys of scale t is given

by

(Qstretch (-2Thus, perturbations in the elasticity field Q grow exponentially (on all scalesfor which the Zimm criterion is satisfied) at the rate V(t)/t C t -2/3. ForKolmogorov turbulence, small scale elasticity field perturbations grow mostrapidly. Of course, for I < td the growth rate drops, as such scales fall intothe hydrodynamic dissipation range. This process of eddy-stretching inducedgrowth of elasticity perturbations continues until the back-reaction of elastic

energy, induced by tension in the polymer strands, becomes comparable tothe mechanical energy of the stretching eddy. Thus, elastic energy growth islimited by equipartition, so that

cvK Q2 poV02((/L) 2 /3. (3- 33)

34

Equation (3-33) assumes 4, < L, where w1/3/f2/3 w,, i.e. the largest acti-

vated scale of Q.

In the ensuing phase of fully developed viscoelastic turbulence, two types

of excitations co-exist in the polymer solution. The first are (low frequency)

fluid eddys, with w(e) -- V(e)/l. The second are (high frequency) elastic

waves, with w(t) - VE1t. Here V/E is the shear elastic wave velocity computed

using Qrms, i.e.

V = CpK Qms/p. (3-34)

As fVE contains contributions from all eddys with I < 4,, V/E > V(t) for

£ < 4,. This state of viscoelastic, polymer hydrodynamic turbulence is closelyanalogous to magnetohydrodynarnic turbulence, the basic properties of whichare summarized in Appendix A. As in the case of MHD turbulence, elastic

waves tend to impede inertial range transfer, since eddy interaction can occuronly when two neighboring elastic waves propagating in opposite directions

collide[14], generating a low frequency virtual mode which interacts with thefluid eddys (see Figure 3-4). Consequently, as in MHD, the inertial range

transfer rate for viscoelastic turbulence is reduced[20], i.e.

ltm) -- V(t) V(f) (3-35)

1VE

Hence, self-similarity of energy transfer implies V(t) = (jVE)1/ 411/ 4 and

E(k) = (47E)1/2k-3/2, (3 - 36)

where Ek , EQ , E(k). Note that the inertial range spectrum of PHDturbulence is -, k-3/2 (as in MHD) rather than the familiar k-' 13 of theKolmogorov theory. Furthermore, the dissipation scale of PHD is also dif-

ferent from that of Kolmogorov turbulence and, following the arguments of

Appendix A, is easily seen to be

4, = (f/C)1/3 V2,l (3 - 37)

Perhaps the most basic question one can ask about PHD grid turbu-

lence is: does adding the polymers increase or decease the rate of energy

35

SWE(k) + WE(k') < < WE

Figure 3-4. Eddy - Elastic Wave Interaction.

36

dissipation?! This question can now be answered by simply computing the

difference of the energy dissipation rates for PHD turbulence and ordinary

hydrodynamic turbulence, i.e. by calculating:

Af= Jd, (vk2Ek(k) + 2w.EQ(k)) (3-38)

- jd vk 2E(k).Jks

Here, Ek and EQ are the (equipartitioned) mechanical and elastic energies

which add to the result given in Equation (3-36), kd, = £d, kd =(0/0

(the Kolmogorov microscale), k_ = tI;1, and E(k) = -2/3k-'/3. Note that the

lower bound of the range of integration is k, in each case, since only those

scales on \vhich the polymers are 'activated' can exhibit altered dynamicsand dissipation rates. Straightforward arithmetic indicates that:

Ac > O (3-39)

so that the net dissipation rate is increased! The key effect in determiningthat Ac > 0 is that Zimm relaxation-mode-induced dissipation is not 'small'

for scales with I < 4. (i.e. see Figure 3-5), as is viscous dissipation. Alter-

natively put, energy must be dissipated in fluid-polymer friction drag duringthe process of stretching the polymer. It should also be mentioned that more

detailed polymer models would consist of a spectrum of Zimm relaxation

modes, each with a corresponding relaxation frequency. In that case, one

might expect increased Zimm dissipation losses, since a broad spectrum of

dissipation modes (rather than just one) would resonate with the fluid eddys.If a sufficient number of relaxation modes were excited the cascade could be

effectively truncated. Finally, we note that the prediction that Ac > 0 shouldbe amenable to experimental examination.

At first glance, the result of Equation (3-39) seems disturbing, since drag

is predicted to increase! However, in interpreting the result one must keep inmind the important differences between grid turbulence and pipe flow turbu-

lence. In particular, the results of the above analysis suggest that in a pipe

flow a viscoelastic buffer layer will form in the region xv < x <x = U./w,of the turbulent boundary layer (see Figure 3-6). Here xv = v/U. is the

37

eg213

--- --- - - --- -- - -- - - - w

Figure 3-5. The Spectrum of Inertial Range Transfer Rates.

38

Core

IIIII

Figure 3-6. The Viscoelastic Boundary Layer.

39

width of the viscous sublayer. Within this viscoelastic buffer layer, all ed-

dys satisfy the Zimm activation criterion, and thus one can expect a mod-

ified inertial range spectrum and, most importantly, enhanced dissipation

in the layer. The enhanced dissipation in turn results in reduced momen-

tum transport, and thus in enhanced slip and drag reduction. We note that

this scenario is, in effect, a variation on the 'Newtonian plug and buffer

layer' scenario of Lumley, with enhanced dissipation entering via W,. rather

than thru augmented viscosity. Unlike the Lumley[7] scenario, our model

does not require the incidence of a coil-stretch transition. It should also be

mentioned that like the model of DeGennes and Tabor[8],[15], our model

predicts an equipartition of turbulent mechanical and elastic energy. It does

not, however, predict a 'truncated' cascade, but rather a modified, viscoelas-

tic cascade. Finally, it should be mentioned that our model fails to account

for the observed anisotropy of inertial range eddys in drag reduced flows and

for the relation of this phenomenon to the observed decorrelation of stream-

wise and cross-stream velocity fluctuations and thus to drag reduction. A

possible resolution of this deficiency may come from the proper treatmentof the small scale turbulence dynamics in the presence of a strongly sheared

mean flow.[25],[26]

40

4 CONCLUSIONS AND RECOMMENDA-TIONS

4.1 Conclusions

In this report, we have surveyed the existing experimental and theoretical

understanding of drag reduction by polymer additives, and have presented a

new continuum model (polymer hydrodynamics - PHD) of turbulent poly-

mer solutions. Several natural extensions of and directions for further devel-

opment of this model are apparent. These include:

1. the application of standard perturbative closure methods (D.I.A., etc.)[27]

to the PHD equations to develop tractable spectrum evolution equa-

tions for PHD turbulence.

2. investigation of PHD turbulence in actual bounded shear flow configu-

rations. This would probably develop as a continuation of 1., above.

3. the generalization of the PHD model to incorporate non-Hookean elas-

tic effects[16].

4. an investigation of the coil-stretch transition in the context of a dy-

namically relevant chaotic or turbulent flow.

It is clear that the study of drag reduction in turbulent polymer hydro-

dynamics is a difficult yet fascinating enterprise!

41

4.2 Program Recommendations

We believe that the time is right for an experimental and theoreticalprogram to understand the detailed mechanics of polymer drag reduction.

Such a program would consist of several parts:

1. Measurement of the velocity and strain rate correlation functions andpolymer properties (extension and orientation) of turbulent polymersolution, which are homogeneous but are not driven at an outer scale;homogeneous turbulence which is driven in a Kolmogorov cascade froman outer scale; and pipe wall turbulence.

2. Measurement of drag, turbulence, and velocity structure in pipe flowsselectively injected with polymers, to determine which effects of poly-mers are important: creation of a buffer layer, modification of the prop-erties of the turbulence in the fully turbulent fluid, or others.

3. An extensive program of numerical simulation of turbulent polymer so-lutions and pipe flows, such as that already underway at the La JollaInstitute. It is also possible to make diagnostic measurements (such asof the Lagrangian strain rate and velocity autocorrelation functions)which would be very difficult in the laboratory. The great flexibilityand versatility of these methods, complementing real laboratory exper-iments, makes them powerful and useful tools.

4. Theory to understand and interpret the results of laboratory and nu-merical experiments.

On account of the difficulty, complexity, and importance of polymer dragreduction phenomena this work will require the level of sophistication in ex-periment and theory which are the state-of-the-art in hydrodynamics, plasmaphysics, continuum mechanics and computational physics. It will need todraw in established professionals from those fields.

42

A BASIC PHENOMENOLOGY OF MHDTURBULENCE

In this Appendix, we summarize the basic phenomenology of MHD tur-

bulence essential to our discussion of polymer-hydrodynamics. The most sig-

nificant feature of MHD turbulence is the modification of the inertial range

spectrum due to the presence of 'large' scale magnetic perturbations. The

effect of such magnetic perturbations is to inhibit inertial range transfer by

coupling small eddys to Alfven waves which propagate along the large scale

magnetic perturbations. This transfer-inhibition is commonly referred to as

the Alfven effect[20].

Here, we present a heuristic derivation of the Alfven effect by exploring

the effect of a uniform magnetic field B = Boi on inertial range transfer.

For the k mode, the MHD equations are:

Ofhk ý(k k _0 (' + .'~V (p + B_ -• +._L/2) (A-1)

Po(-

and ak V 4)kBo- - ýV_. + (z& • V k)k (A -2)at +kz

Here, the inertial range transfer rate for the eddy of scale .j1-k is just 1/'_ =

(i. Y k)., which we estimate using standard iterative closure methods, i.e.:

1/7ýkA k-• k, fl)(A -3)

where:

j/(2 ig MkZB0 - (2)=A ~. V = Y-k' 4kfkP :

6(2) -(2)A1Wk" ,A+kI = Boik, Los,. (A-4)

Here, the only virtual velocity-velocity interactions are retained, since we

only seek to determine the effect of B0 on fluid eddy-shearing. Furthermore,

for simplicity, a single parallel wave vector is assumed. Here, Awk.. is the

43

inverse decorrelation time for the k + k' virtual mode. Combining Equations

(A-2) and (A-3) yields:

(k.•- (A - 5)21/# kAWk" 2

Here VA = Bo/Výp is the Alfven velocity. Enforcing locality of transfer (i.e.

k k_') then yields:

A)! \2) 1/21/1kq k•J/1 _k_ kJ " (A -6)

In the limit VA -- 0 (i.e. no magnetization), Equation (A-6) reduces to the

familiar relation for Kolmogorov turbulence, i.e. 1/71k , kl~k. In the strong

magnetization limit, k,IVA > Awk", Equation (A-6) implies that

I/ k A~k (A - 7)

It is apparent that a strong magnetic field (i.e. k5VA > Aw) acts to reduce

inertial range transfer. Interpreting Equation (A-7) iteratively then yields:

k TA _)_ (A-8)

Equation (A-8) states that a strong magnetic field inhibits inertial rangetransfer by increasing eddy lifetimes by a factor of kVA/kI4k.

The result of Equation (A-8) is appropriate to the case of a DC magneticfield. Since large scale magnetic fluctuations will behave similarly, and sinceAlfvenic interaction is non-local in k-space (note that the I x B force cannotbe eliminated by a Galilean transformation!), we can generalize Equation (A-8) by arguing the correspondence kVA -* kV', where VA = (Ji)rms/vf-,the Alfven velocity calculated using the root-mean-square fluctuating fieldfirms, which is dominated by contributions from the large scale magneticperturbations. In that case:

1/Trk = (kfk) 2/kVA, (A -9)

44

which is the well-known but heretofore unjustified result for the Alfvenically

modified eddy turn-over rate appropriate to MHD turbulence.

The notion that magnetic fields inhibit inertial range interaction is fre-

quently expressed by the alternative statement that two eddys will riot inter-act unless they propagate in opposite directions along neighboring magneticfield lines (i.e. see Figure 3-5)[161. This conception of the Alfven effect can bemanifested in the simple model presented by relaxing the simplifying assump-tion of a single k.. In that case, k, -* k,+kk' in Equation (A-6), which in turn

implies that energy transfer will be inhibited unless (k, + k,)VA > Awk-. Of

course, (k, + k' )VA -. 0 is mathematically equivalent to the physical picture

that two Alfven waves propagate in opposite directions. It is also interest-ing to note that the 'inhibition factor' (1 + (k' kA)2/Aw_,,) 1'/ 2 suggests ananalogy between inertial range intemction and nonlinear Landau damping ina plasma[28], with fluid eddys as the analogue of particles and Awk,, as thecounterpart to the spectrum-averaged ballistic frequency (Ak)Vth, where Vth

is the particle thermal velocity. From that view-point, the inhibition factorimplies that wave-eddy interaction occurs when k.. VA < AW-, just as non-linear Landau damping occurs when two waves beat to drive a 'virtual' modewhich resonates with the particles, i.e. when w + w' = w" < AkVth. Alter-

natively, just as spectral transfer due to nonlinear Landau damping is smallin the ratio (Ak)Vth/w, inertial range transfer in MHD turbulence must besmall in the ratio Awt/klkVA. In the corresponding case without a DC field,Aw/kZVA - -Vf(t)/i'A, thus recovering Equation (A-8).

Having derived the effective eddy turn-over rate for MHD turbulence, it isstraightforward to determine the modified inertial range spectrum. Balancing

the dissipation rate c (which is also the inertial range energy flow-thru rate)with V(t) 2/t(t) (where V(I) and r(e) are the velocity and effective turn-over

rate for an eddy of scale 1) yields:

V=()2 V()2V(1) TA(M) (A-l0)Tr(t) r T(t)

V(1)4IVA

45

Thus V(f) = (VA)I/411/' and hence

E(k) - (IA)1/2k-3/2. (A - 11)

Thus, the Alfven effect changes the inertial range spectrum from E(k) =

f2/ 3 k- 5/3 to E(k) = (4VA)1/2k-3/2. Note that since rA(f) < r(f), inertial

range mechanical and magnetic energies are equal. Similarly, the modified

dissipation scale Id can be determined by balancing the modified eddy turn-

over rate rWt)- with the viscous dissipation rate v/1 2, where v is the kine-

matic viscosity. This implies:

V V(fd) 2

F = A- 12)

so that using the relation V(Id) = (vA)y/4eY/ 4 yields

Id - v2/3(f/E)1/3, (A - 13)

which is the dissipation scale for MHD turbulence. Note that the assumption

of v (. 7 (where q is the resistivity) is implicit in this analysis.

46

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Christensen, J. Fluid Mech. 30, 305 (1967)

3. P. S. Virk, Al. Ch. E. Journal, 21 624 (1975).

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17. N. Wax, ed. Selected Papers on Noise and Stochastic Processor (Dover,

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18. P. G. DeGennes, The Physics of Liquid Crystals, (Oxford, New York).

19. H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids(Cambridge, New York, 1978).

20. R. H. Kraichnean, Phys. Fluids 8, 1385 (1965).

21. P. S. Iroshnikov, Soy. Astron. Y, 548 (1964).

22. G. K. Batchelor, Proc. Roy. Soc. A 2a1 349 (1950).

23. P. G. Saffman, J. Fluid Mech. 16 545 (1963).

24. R. H. Kraichnan and S. Nagarajan, Phys. Fluids 10, 859 (1967).

25. D. Biskamp, Phys. Fl. B1, 1964 (1989).

26. T. Chiueh, P.. Terry, P. H. Diamond, J. E. Sedlak, Phys. Fl. 29, 231(1986).

27. H. Biglari, P.H. Diamond, P. W. Terry, Phys. Fluids B2, 1 (1990)

28. R. H. Kraichnan, J. Fluid Mech. 5 497 (1959).

29. R. Z. Sagdeev and A. A. Galeev, Nonline Plasma Theory, (Addison-Wesley, Menlo-Park, 1969).

48

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